Fractal Plate Reconstruction?

Fractal Plate Reconstructions, Seafloor Spreading Asymmetry, and Kinematics utilizing Fractal Criteria

Summary

Information theory and fractal analysis provide a basis for a new approach to plate reconstructions of magnetic isochrons and fracture zone crossings. The new approach models accretionary boundaries as fractals (via variable binning sizes) and seeks parameter sets that produce a minimum sum of logarithms of bin numbers. Spline interpolation of a trial reconstruction parameter set, including spreading asymmetry along ridge segments, and Monte Carlo modifications to the parameters provide iterative calculation of the full fractal set. Such a method allows for incorporation of all isochron identifications from the plates of interest in a single, iterative calculation set rather than just a few separate, selected individual reconstruction calculations. The splined parameter sets also provide a means for calculating instantaneous rotation rates and asymmetries. More detailed analysis of the fractal results for a number of the “best” solutions also provides alternative approaches to characterization of the uncertainty in the derived reconstruction parameters; more work will be required in this area.

Introduction

Plate Reconstructions and Kinematics

Considerable progress in plate reconstruction techniques, especially incorporating uncertainties, is manifest in improved, higher resolution rotational parameter sets for the world oceans (e.g., sources used by Müller et al., 2008). In most cases, plate reconstruction sets are calculated discretely for specific, corresponding magnetic isochrons on complementary plate pairs (e.g., applying the methods of Kirkwood et al., 2002, following Hellinger’s, 1981, great circle parameterizations of accretionary and transform plate boundaries). However, interpolated plate kinematic calculations and global reconstructions, via constant finite difference poles and rates (e.g., Pitman & Talwani, 1972) or via spline interpolation of pseudovectors (Pilger, 2003), do not provide intrinsic uncertainty estimates (e.g., Hanna and Chang, 2000). Further, fitting criteria for reconstructions typically include seemingly necessary, but ad hoc assumptions about the configuration of accretionary plate boundaries, requiring additional “nuisance” parameters (e.g., the poles to the multiple great circles of Hellinger, 1978; see also Kirkwood et al., 2002) in order to characterize boundary shapes, thereby complicating uncertainty characterization.

Alternative approaches to plate reconstructions which do not require nuisance parameters include that of McKenzie et al., (1970), minimizing overlap area squared, and Pilger (1978), minimizing normalized area-squared. Unfortunately, neither provides an obvious means of estimating uncertainties. The motivation for the current study is to propose a further alternative criterion for optimal plate reconstruction that does not require a large number of nuisance parameters and has the future potential for uncertainty characterization of both finite and instantaneous kinematic parameters.

Fractals

Empirical Basis

Mandelbrot’s (1975) fractals have become an empirical cottage industry over the past three and one-half decades, with subsidiaries in most of the hard and soft sciences. Within the earth sciences the Richter-Gutenberg earthquake magnitude scale, minted years before “fractal” was coined, is perhaps the most familiar example of self-similar structure across a spectrum of scales. Coastline lengths (Richardson, 1951, Mandelbrot, 1967), manganese oxide dendrite shapes (Chopard et al., 1991; Merdan and Bayirli, 2005), fault-fracture extents (Okubo and Aki, 1987; Aviles et al., 1987, Walsh and Watterson, 1993), alluvial drainage patterns (Rodriguez-Itur et al., 2001), and tectonic plate areas (Bird, 2003; Sornette and Pisarenko, 2002) are just a few of the additional diverse types of geophenomena that can be identified as fractal. Even earlier, turbulence may well have been recognized as “fractal”: Richardson’s (1922 p. 66), poem, a takeoff on Jonathan Swift’s Big fleas and little fleas: “big whirls have little whirls that feed on their velocity, Little whirls have lesser whirls. And so on to viscosity.” Perhaps most relevant to plate boundary configurations are observations that trace lengths of fault and fracture sets observed in outcrop and in subsurface and reflection seismic sections demonstrate fractal distributions (as referenced above). Could divergent plate boundaries, by their very nature comprised of faults and fracture systems, exhibit fractal behavior?

Rationale
One of the catalysts for Mandelbrot’s initial conception (see, e.g., 1953) of self-similarity was Shannon’s (1948) information theory and its application to human languages (1951). After exploring self-similarity in economics, fractals as an extra-humanities concept emerged with the unearthing of the study of the indeterminate length of the coastline of England (Richardson, 1951, published in 1961, and cited by Mandelbrot, 1967). The measured coastline length increases as the size of the measuring stick decreases; plots of the logarithm of coastline length versus the logarithm of the measure produce linear trends with negative slope for numerous analyzed coastlines, although the calculated slopes of the log-log plots vary from island to island and continental oceanic frontier to frontier. Feder (1988) has shown that box (or bin) counting (the number of boxes intersected by a coastline) for grids of various sizes produces comparable fractal measures to the measuring-stick method.
While fractals have been found to occur progressively “everywhere,” (in both mathematical investigations and natural observations, e.g., Schroeder, 1991, Barnsley, 1993), a foundational physical or stochastic rationale for their occurrence beyond what Mandelbrot (1953) provided before introducing “fractals,” and which he and others gave in mathematical elaboration of fractal structures, was largely overlooked until the last few years of the Twentieth Century. Such a theoretical underpinning was provided in 1998, as Pastor-Satorras and Wagensberg, returning to information theory roots, especially Jaynes’ (1957) extension of Shannon’s work, demonstrated that fractals can be interpreted as realizations of processes that maximize information (Shannon) entropy over a large range of scales with respect to a particular set of information. By maximizing entropy the most probable configuration (or distribution) of a set of essentially identical “atoms” (or information kernels) of a system is achieved, subject to available constraints (Jaynes, 1957; see also Grandy, 2008). Further elaboration of the importance of such a state is provided below.
Accretionary Plate Boundaries: Fractal?
Are plate boundaries, specifically spreading centers and transform faults, fractal? And, if they are, is there any significance and/or value in their possession of such structures? As a preliminary answer to the question, a digital representation of the principal contemporary plate boundaries, compiled by Bird (2003) was subjected to analysis. Bird’s extensive compilation provided latitude-longitude sets for each plate boundary, identified by type (including vergence direction for convergent boundaries)and velocity, where possible.
From Bird’s (2003) larger data set, spreading center boundaries separating the oceanic parts of the Eurasian, North American, African, South American, Indian, Antarctic, Australian, Pacific, Nazca, Juan de Fuca, and Cocos plates were abstracted. The resulting data set consists of both divergent and shear boundary segments without differentiation, with data point spacing of no more than 111 km.
Two approaches to fractal analysis of the abstracted data were undertaken. In the first, the approximately 2200 data points were converted from spherical (latitude and longitude) to Cartesian coordinates with unit norm and displaced to a center point at (1, 1, 1).
For a particular cubic grid spacing, Du_m, each data point (x_n, y_n, z_n) was assigned to a particular three dimensional bin (i, j, k), by the formulas: i = (integer) (x_n/Du_m); j = (integer) (y_n/Du_m); k = (integer) (z_n/Du_m). Then, for each grid spacing, Du_m, the number of bins, N_m, with one or more data points was tabulated, producing the set (Du_m, N_m). A plot of Du versus N for the divergent plate boundary set is shown in Fig. 1. As expected, a larger number of bins is occupied at finer grid spacings, with, conversely, the number of bins decreasing with increasing bin spacing.

Figure 1. Bin count versus spacing for principal spreading center boundaries.

By computing logarithms (to base 10) of each coordinate, additional structure is apparent in the plot (Fig. 2). A steeply inclined log-log-linear trend is apparent for spacings greater than approximately 2.7-2.8 (~450-600 km). Another log-log-linear trend, with a lesser slope is apparent between approximately 1.8 (~60 km) and 2.7. For values less than 1.6-1.7 log-log-linear trends are not apparent. The presence of a log-log-linear trend is the leit motiv of statistical fractal structure (and its implied maximum entropy foundation).

Figure 2a. Log (count) versus log (spacing) as in Fig. 1.

Figure 2b. Linear curves fit to log-log plot of Fig. 2a.

Trend lines fit to the two log-log trends produce slopes of approximately -1.25 and -0.081 (Fig. 3). It’s tempting to associate the boundaries between trends, especially the two log-log-linear curves, with depth to the transition zone, based on some sublithospheric convection models of the asthenosphere; that is, vertical and horizontal cross section convection cell scales are on the same order (e.g., Ballmer et al., 2007). Whether it is reasonable to make such an inference is not the primary object of this contribution. Rather, the apparent fractal nature of divergent plate boundaries, along with the foundation of fractals in information theory, provides a rationale for an alternative approach to plate reconstructions.

The initial fractal gridding procedure relied upon three-dimensional binning of data points which are essentially restricted to the surface of a near-sphere. Whether such a parameterization is appropriate might be questioned. As an alternative, the second approach applies the original coastline measurement scheme of Richardson (1951; see also Mandelbrot, 1967, 1976) might be more applicable. In this case, each “length” of each discrete seafloor spreading boundary from Bird’s (2003) data set was calculated over a range of length intervals of the “measuring stick,” without distinguishing between spreading ridge or transform fault segments. Given the set of data points, (x_n, y_n, z_n), for each spreading ridge, and beginning with the first point, n=0, and great circle angle increment Da_i, the angle, a_n, between the starting point (x₀, y₀, z₀) and each successive point (x_m, y_m, z_m) was calculated until a_m ≤ Da_i ≤ a_m+1. Then the (x,y,z) point along the great circle connecting (x_m, y_m, z_m) and (x_m+1, y_m+1, z_m+1) which is exactly (to machine error) Da_i from (x₀, y₀, z₀) was determined. The resulting point became the new origin for the next calculation, while the net angular “length” was incremented by Da_i. Once the full length of the spreading ridge was “measured” by whole angular intervals, the fractional amount from the last calculated origin to the last data point of the ridge was added to the total measured length for the current Da_i. The procedure was repeated for each desired Da_i, and the results tabulated for each spreading boundary.

The results of these calculations (raw plot in Fig. 3a), in Fig. 3b (on a log-log scale) produce the familiar log-log-linear trend of fractals for each spreading ridge. (The Nazca-Antarctic ridge produces two anomalies: a stair step effect at large angular spacing, representing the distinctive offsets of the ridge by transform faults, and a flattening at small intervals, representing coarser sampling of the ridge along a great circle segment. A slight stair step effect is also apparent in the Pacific-Cocos ridge at larger angular spacings, and also attributed to the transform fault offsets of the ridge.)

Figure 3a. Bin count versus measure spacing for principal spreading centers.

Figure 3b. Log (bin count) versus log (measure spacing) for principal spreading centers.

A tentative conclusion of these analyses is that spreading ridge segments manifest approximate statistical fractal structure. Power series fits to the main fractal segments imply fractal dimensions of each on the order of 1.1 +/- 0.1 for each ridge segment.

Fractal Reconstruction Formalism

Data and Initial Parameters

Plate reconstructions on a sphere, because they involve finite rotations, are inherently non-linear; thus any effective reconstruction algorithm is iterative (e.g., Pilger, 1978; Hellinger, 1981; Chang et al., 2000). To illustrate the fractal plate reconstruction method, a single plate pair (Australia-Antarctica) was selected, and data from the contemporary ridge location, identified magnetic isochrons, and identified and dated fracture zone crossings were assembled (digitized from the map identifications of Cande & Stock, 2004), together with preliminary reconstruction parameters over the range in age of the isochrons.

Continuous Parameterization

As a starting point for the fractal reconstruction analysis, the finite rotation parameter set (age: t, latitude: f, longitude: q, rotation angle: l) of Cande & Stock (2004) for Australia-Antarctica which they derived for their interpreted isochrons and fracture zone crossings, was converted to pseudovectors with magnitude equal to the average rotation rate (rotation angle l_i for age t_i, in Ma, divided by t_i; note that parameterization using rotation rate magnitude allows for spline interpolation of three dimensions versus age). For time zero, the rate is the instantaneous, contemporary rotation rate, and the instantaneous pole parameters are also applied initially (DeMets et al., 1994; incorporation of the more recent model of DeMets et al., 2010, would have negligible effect in this particular exercise).

x_i = (l_i / t_i ) cos f_i cos q_i (eq. 1)

y_i = (l_i / t_i ) cos f_i sin q_i (eq. 2)

z_i = (l_i / t_i ) sin f_i (eq. 3) Each Cartesian parameter set, (e.g.: x_i, i=1,n), with corresponding ages (t_i, i=1,n), was then separately interpolated using a cubic spline parameterization (x, y, and z versus t, using the algorithm of Press et al., 1998, as outlined by Pilger, 2003) at intervals of 5 m.y.; the resulting interpolated data were then re-splined to produce the initial model. Asymmetry factors for ridge segments that have data points (initially equal to 0.5) were also spline interpolated at the same time interval (5 m.y.)
Aside: The interpolation of the pseudovectors to a regular time interval could be viewed as an unnecessary complication in the current application as the data set consists of magnetic chrons and fracture zone crossings of which each corresponds in age with Cande and Stock’s (2004) reconstruction parameters; the alternative would be to solve for parameters for the age of each identified chron. However, should this method be applied to a much larger data set with a wide variety of ages, a regular interpolation interval would be more appropriate, particular if uncertainties were incorporated.

The digitized isochrons were visually assigned to distinct spreading segments bounded by active or paleo- transform faults to allow for variation in spreading asymmetry. For each age tm (of the dated magnetic isochron and fracture zone crossings), the spline parameters produced interpolated xm, ym, and zm, the pseudovector rotation parameters. The equivalent rotation parameters were determined from xm, ym, and zm by the familiar inverse transformations: q_m = tan^-1 (y_m / x_m); f_m = sin^-1 [z_m /(x_m² + y_m² + z_m²) ^½]; l_m = t_m (x_m² + y_m² + z_m²) ^½. Each data point was then restored to a trial “ridge” location by the negative rotation angle multiplied by the interpolated asymmetry factor (initially, 0.5) for the ridge segment to which the data point was assigned.

From the resulting coordinates of the rotated points, (f_d, q_d) and for each cubic bin dimension, Df_k, the data points were assigned to bins (with coordinate indices m, n). All bins for a particular Df_k (equal to the latitudinal dimension) have the same area. The longitudinal dimension, Dq_k, is determined by:

Dq = (Df_k)² / [1+sin(f₀) - sin(f₁)] (eq. 4),

in which

f₀ = [(integer)(f_d / Df_k)] Df_k (eq. 5),

and

f₁ = f₀ + Df_k (eq. 6).

The bin indices are, then,

n = (integer) (f_d / Df_k) (eq. 7)

and

m = (integer) (q_d / Dq_k) (eq. 8).

Once all data points were rotated and binned, the number of bins occupied for each bin dimension was tabulated (with isochron data points tabulated separately from fracture zone crossings). Note that this fractal bin-counting algorithm is two dimensional (latitude, longitude) as opposed to the three-dimensional bin-counting for contemporary divergent and shear plate boundaries described above (the first fractal binning application). Also, note that the longitudinal boundaries of each bin obviously vary with latitude.

Iterative Improved Parameterization
It is postulated that improved fits of rotated isochron and fracture zone crossings can be achieved by reducing the sum of the bin counts across the full range of bin spacings.
To improve the fit of the rotated isochrons and fracture zone crossings, it is necessary to reduce the integrated bin-counts. One could create a non-linear fitting algorithm that incorporates the numerically interpolated gradient of the bin counts relative to all of the parameters. Alternatively, a Monte Carlo approach was be applied.
The latter approach was undertaken as follows:

From the Cartesian parameterization, the rotation parameters for iteration k were modified by the addition of a random factor, rk:

x_ik = x_i0 (1 + r₁) (eq. 9),

y_ik = y_i0 (1 + r₂) (eq. 10),

z_ik = z_i0 (1 + r₃) (eq. 11),
and

a_ijk = a_ij0 (1 + r₄) (eq. 12).

Each r_k is a randomly generated number (e.g., Press et al., 1998), between -0.5 and 0.5, multiplied by a scale factor (e.g., 0.01), and different for each coordinate (including asymmetry, aijk), each age, each segment (for the asymmetry factor), and each iteration.

For each Monte Carlo realization, the sum of the logarithms of the bin values was calculated for the full range of bin sizes. The minimum sum for a set of iterations produced the most compact log-binning of the rotated isochrons and crossings among the realizations. (One could sum the bin values directly, rather than the logarithms, giving greater weight to the smaller bin values.)

For each Monte Carlo iteration, the associated rotation parameter sets were stored and indexed. That set which produces the minimum summed logarithms of bin values (for both chron and fracture zone data) was then identified.

Application

As indicated above, magnetic isochrons and fracture zone identifications from the Southeastern Indian Ocean, Antarctic and Australian plates, as interpreted and published by Cande and Stock (2004; their Fig. 2), were digitized utilizing the GeoGraphix® Discovery™ software. Isochron ages were assigned according to the geomagnetic time scale of Gradstein et al. (2004).

Beginning with the published parameters of Cande & Stock (2004), the parameters were interpolated using cubic splines at 5 m.y. intervals from 5 to 35 Ma. The iterative Monte Carlo procedure outlined above was followed, with isochrons and fracture zones binning initially treated separately, then combined to determine the minimum sum of the logarithm of the bin counts. 40,000 iterations were performed.

Fig. 4a illustrates the fractal counts of the initial reconstructed isochrons and fracture zones. Fig. 4b illustrates the same bin counts in log-log plots. Note the near linear trend for larger bin spacing.

Figure 4a. Fractal counts of half-angle reconstructed isochrons and fracture zone crossings, Australian-Antarctic Ridge, based on initial spline-interpolated parameters.

Figure 4b. log(bin count) versus log (spacing) as in Fig. 4a.

Fig. 5 displays the same curves as Fig. 4 for the initial calculations and, in addition, corresponding curves which produced the minimum bin counts for the combined reconstructed chron and fracture zone crossing data. Note how most bin counts for the reconstructed chrons are less than the initial calculations. In contrast, note that there is little change in the calculated bin counts for the reconstructed fracture zone crossings. Thus, most of the change came from the reconstructed chrons.

Figure 5a. Fractal counts of half-angle reconstructed isochrons and fracture zone crossings, Australian-Antarctic Ridge, based on final spline-interpolated parameters.

Figure 5b. log(bin count) versus log (spacing) as in Fig. 5a.

Table 1 provides a comparison of the initially interpolated and final parameters, including differences. Fig. 6 illustrates the originally interpolated and modified (minimum summed logarithms of the bins) parameters and final differences. Table 2 provides the “final” modified asymmetry factors for each spreading segment and isochron age.

Figure 6a. Initial and final total reconstruction parameters, interpolated at 5 m.y. intervals.

Figure 6b. Differences between original and final (minimum log fractal bins) spline interpolated parameters.

Fig. 7 illustrates the smallest and largest bins for the initial reconstruction parameters. Fig. 7b, zoomed in on a central portion of the Australian-Antarctic ridge, illustrates the reconstructed data points; note how each data point is associated with a bin (some data points are associated with the same bin).

Figure 7a. Largest and smallest fractal bins "covering" reconstructed data points for initial interpolated parameters, assuming symmetrical spreading. Also indicated are segments of the spreading center for which asymmetry factors are calculated.

Figure 7b. Zoomed-in fractal bins, as in Fig. 7a, together with initial reconstructed data points. Note how each of the smallest bins is associated with individual points.

Figure 8a. Reconstructed Australian-Antarctic Ridge, based on initial parameters with symmetrical seafloor spreading. (Click to enlarge.)

Figure 8b. Close-up of eastern ridge segment in Fig. 8a. Note the primary two zones of reconstructed points, indicating either asymmetric spreading or ridge jumping. 

 Figure 8c. Close-up of central ridge segments in Fig. 8a. Note the scattering of restored data points.

Figure. 9a. Reconstructed Australian-Antarctic Ridge, based on final parameters with asymmetrical seafloor spreading.

Figure 9b. Close-up of eastern ridge segment in Fig. 9a.

Figure 9c. Close-up of central ridge segments in Fig. 9a. Note the greater concentration of data points by incorporation of asymmetric spreading (which, alternatively, may involve discrete ridge jumps).

Figure 10a. Reconstructed Australian-Antarctic Ridge, based on initial (with symmetrical spreading) and final (with asymmetrical spreading) parameters. Symbol colors as in Fig. 6 and 7 (final symbols are posted on top of initial).

Figure 10b. Close-up of eastern ridge segment in Fig. 10a. Note the improved concentration of data points by incorporation of asymmetric seafloor spreading.

Figure 10c. Close-up of central ridge segments in Fig. 10a.

Discussion

The purpose of this contribution is to introduce an alternative approach to iterative plate reconstruction parameterization. One of the proposed advantages of this new approach is the ability to incorporate a wide variety of isochron identifications, not just those for which a few finite reconstructions are to be calculated. Further, it demonstrates the viability of incorporating asymmetric spreading factors for spreading segments as additional parameters, in this example, modeled by the Monte Carlo calculations.

For several reasons, only a limited number of iterations of the Monte Carlo realizations for the Indian Ocean example was run: (1) Likely digitization errors of the isochron and fracture zone crossings provide additional uncertainty, so that the “improved” parameters in this study should not be viewed as superior to the published parameters of Cande and Stock (2004). (2) Only selected isochrons crossings are available for this analysis from the published work. Ideally, all isochron identifications could be used in this integrated approach. (3) As the Monte Carlo simulations proceeded, parameter sets which produced improved fractal fits were fewer in number.

It is desirable for full sets of isochron and fracture zone crossing identifications from each of the ocean basins to be subjected to this approach, including asymmetric spreading factors. It is also possible to develop alternative binning criteria (e.g., rotating data sets to the equator so as to minimize bin “deformation”). And, there is no reason, beyond computer resources, why data sets which share one or more triple plate accretionary junctions couldn’t be subjected to simultaneous modeling for reconstruction parameters. Finally, there are alternative strategies to Monte Carlo simultions that might be pursued for achieving optimal fits. Since fractal binning is intrinsically discontinuous, it is not possible to calculate analytic derivatives, so non-linear least-squares approaches are not possible. Exploration of alternative approaches in this area is underway.

The meaning of fractal optimization
Pastor-Satorras and Wagensberg's (1998) insight into the connection between information theory and fractals, that fractals maximize information entropy across a range of scales relative to a kernel of information, is exploited here in a seemingly paradoxical way. That is, the numerically integrated logarithm of the fractal count is minimized by varying the rotation parameters (including segmented asymmetry) to produce the maximum entropy relative to the optimal rotation parameters (the kernel of information). So, the fractal bin distribution incorporates both the underlying stochastic process of seafloor spreading as well as errors in data identification, location, and digitization.  

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©2010 Rex H Pilger Jr